A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).
1, 7, 69, 733, 8061, 90462, 1028871, 11814376, 136643085, 1589311381, 18569375114, 217773347502, 2561944357311, 30219704365104, 357278540928168, 4232449819704768, 50227362114232109, 596988743410929087, 7105534815529752831, 84678089652554263155, 1010268312800732117946
Offset: 0
Programs
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Mathematica
Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(5*n, k));
Formula
a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025